Calculus of Variations and Geometric Measure Theory

J. A. Bárcena-Petisco - M. Cavalcante - G. M. Coclite - N. De Nitti - E. Zuazua

Control of hyperbolic and parabolic equations on networks and singular limits

created by denitti on 23 May 2021
modified on 19 Nov 2022


Submitted Paper

Inserted: 23 may 2021
Last Updated: 19 nov 2022

Year: 2021


We study the controllability properties of the transport equation and of parabolic equations posed on a tree. Using a control localized on the exterior nodes, we prove that the hyperbolic and the parabolic systems are null-controllable. The hyperbolic proof relies on the method of characteristics, the parabolic one on duality arguments and Carleman inequalities. We also show that the parabolic system may not be controllable if we do not act on all exterior vertices because of symmetries. Moreover, we estimate the cost of the null-controllability of transport-diffusion equations with diffusivity $\varepsilon>0$ and study its asymptotic behavior when $\varepsilon \to 0^+$. We prove that the cost of the controllability decays for a time sufficiently large and explodes for short times. This is done by duality arguments allowing to reduce the problem to obtain observability estimates which depend on the viscosity parameter. These are derived by using Agmon and Carleman inequalities.